# Simply connected quasi-projective varieties in positive characteristic

I am looking for examples of non-projective (quasi-projective) varieties $X$ defined over a field of positive characteristic, which have trivial étale fundamental group.

It is well known that the étale fundamental group in positive characteristics is a very difficult object, especially so in the non-projective case due to possibly wild ramification at infinity. I'm not even sure if there are examples of the kind above. Is this known?

• Note that any projective variety is quasi-projective. A comparison theorem of Grothendieck gives many examples of projective varieties in positive characteristic with trivial etale fundamental group. So probably you wish to amend your question slightly? Feb 22, 2010 at 11:59
• I am fairly sure that removing a codimension $2$ subvariety from a projective variety doesn't change the fundamental group. So I can take any projective example of dimension $2$ or greater and make it nonprojective by yanking out a point. Feb 22, 2010 at 12:21
• In fact David's comment is SGA1 Corollary X.3.3 and has now reminded me of an old question of mine at mathoverflow.net/questions/5375/… Feb 22, 2010 at 12:29
• @DS: Grothendieck's specialization theorem applies to any projective variety which lifts to characteristic $0$. So for instance $\mathbb{P}^2$ is simply connected in all characteristics. So based on what you say (which I haven't seen before but sounds good to me), $\mathbb{P}^2$ minus a point is an example. Feb 22, 2010 at 12:31
• Technical point: I should have asked for the bigger variety to be regular. Feb 22, 2010 at 12:48

This is an answer to Pete's question on simply connected affine varieties (I can not put it in a comment because of space limitation).

I think that in positive characteristic $p$, no affine irreducible variety $X$ of positive dimension is simply connected. We can assume $X=\operatorname{Spec}(A)$ integral because $\pi_1$ is insensible to nipotent elements (SGA IX.4.10). Let $k[t_1,\ldots, t_d] \subseteq A$ be a finite extension with minimal degree $[k(A):k(t_1,\ldots, t_d)]$. Consider the étale cover $Y\to \mathbb A^d_k= \operatorname{Spec}(k[t_1,\ldots, t_d])$ defined by $s^p-s=t_1$. Then $X\times_{\mathbb A^d_k} Y\to X$ is an étale cover of degree $p$. As $k(Y)$ and $k(X)$ are linearly disjoint over $k(t_1,\ldots, t_d)$ ($k(Y)$ is Galois over $k({\bf t}):=k(t_1,\ldots, t_d)$ and $k(Y)\cap k(X)=k({\bf t})$), the tensor product $k(Y)\otimes_{k({\bf t})} k(X)$ is a field. This implies that $X\times_{\mathbb A^d_k} Y$ is connected.

Here's a remark. It's a generalisation of a theorem of Katz-Lang given by Szamuely and Spieß that for a quasi-projective variety over a perfect field $k$, the abelianised tame fundamental group sits in the following exact sequence

$0\rightarrow T \rightarrow \pi_1^{t,ab}(X) \rightarrow T(Alb_X)\rightarrow 0$

where $T$ is a group related to the torsion subgroup in the Neron-Severi group $NS(X)$ and $T(Alb_X)$ is the full Tate module of the generalised Albanese variety of $X$. The definition of tame covers tries to control exactly this wild ramification at infinity (i.e control the function field for the points in $\mathfrak{X}\setminus X$ where $\mathfrak{X}$ is the corresponding projective). You can find more in Szamuely's new book for example.

It is a direct consequence of Abhyankar's Conjecture (which was proved by Raynaud and Harbater) that if $$k$$ is an algebraically closed field of characteristic $$p > 0$$, then no affine curve $$X_{/k}$$ has trivial etale fundamental group. (Note that for curves, affine = quasi-projective, non-projective, by Riemann-Roch.)

I have some lecture notes on this subject from years back:

http://alpha.math.uga.edu/~pete/fundincharp.pdf

(In contrast to what it says on the first page, they are from 2002.)

Addendum: The comments above give plenty of examples of non-projective quasi-projective varieties with trivial etale fundamental group in characteristic $$p$$ (or really in characteristic quelconque). An interesting question left open by these examples is whether there are any (nontrivial) simply connected affine varieties in characteristic $$p$$. As I have said, the answer is "no" in dimension one.

• Regarding the question of finding a simply connected affine variety (of positive dimension): I'm tempted to say that it is not possible. Embed $X$ into $\mathbb{A}^n$ and take a nontrivial cover of $\mathbb{A}^n$ to get a nontrivial cover of $X$. The gap in this argument, of course, is that the cover of $X$ might not be connected. Any idea how to get around this? Feb 22, 2010 at 14:36
• If it is an open subset if $\mathbb{A}^n$, then they are birational and you can use extensions of $\mathbb{A}^n$ that introduce a common nontrivial extension of their function fields. Feb 22, 2010 at 14:50

$\mathbb{P}_k^1$, over an algebraically closed field, is such an example. You can adapt the proof that $\mathbb{Q}$ has no unramified extensions to show that $k(t)$ has no unramified extensions.

• Hi David, I am looking for non-projective examples. I'll edit the question to make that clearer.
– Lars
Feb 22, 2010 at 12:00